Abstract
Symmetry is a powerful framework to perceive and predict the physical world. The structure of materials is described by a combination of rotations, rotation-inversions and translational symmetries. By recognizing the reversal of static structural rotations between clockwise and counterclockwise directions as a distinct symmetry operation, here we show that there are many more structural symmetries than are currently recognized in right- or left-handed helices, spirals, and in antidistorted structures composed equally of rotations of both handedness. For example, we show that many antidistorted perovskites possess twice the number of symmetry elements as conventionally identified. These new ‘roto’ symmetries predict new forms for ‘roto’ properties that relate to static rotations, such as rotoelectricity, piezorotation, and rotomagnetism. They enable a symmetry-based search for new phenomena, such as multiferroicity involving a coupling of spins, electric polarization and static rotations. This work is relevant to structure–property relationships in all materials and structures with static rotations.
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Acknowledgements
The authors acknowledge financial support from the National Science Foundation through the MRSEC program DMR-0820404 and grant DMR-0908718. Discussions with C. J. Fennie, K. M. Rabe and A. M. Glazer are gratefully acknowledged. We thank S. Denev for help with artwork for Figure 4.
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V.G. conceived the idea of rotation-reversal symmetry, the roto groups, their influence on properties, and derived the symmetries listed using symmetry diagrams. D.B.L. critiqued these ideas, helped develop formal definitions for these concepts, and derived the symmetries listed using group theoretical methods. V.G. and D.B.L. co-wrote the article.
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Gopalan, V., Litvin, D. Rotation-reversal symmetries in crystals and handed structures. Nature Mater 10, 376–381 (2011). https://doi.org/10.1038/nmat2987
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DOI: https://doi.org/10.1038/nmat2987
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